> [!NOTE] Theorem > Let $\mu_{X},\mu_{Y}\in \mathbb{R}$ and $\sigma_{X},\sigma_{Y}>0.$ Let $X$ and $Y$ be [[Independence of Two Real-Valued Random Variables|independent real-valued random variables]] such that $X\sim\mathcal{N}(\mu_{X},\sigma_{X}^{2})$ and $Y\sim\mathcal{N}(\mu_{Y},\sigma_{Y}^{2})$ where $\mathcal{N}(\mu, \sigma^{2})$ denotes a [[Normal Distribution|normal distribution]] with parameter $\mu$ and $\sigma^{2}.$ Then $X+Y \sim \mathcal{N}(\mu_{X}+\mu_{Y},\sigma_{X}^{2}+\sigma^{2}_{Y}).$ **Proof**: Since $X-\mu_{X}\sim\mathcal{N}(0,\sigma_{X}^{2})$ and $Y-\mu_{Y}\sim\mathcal{N}(0,\sigma^{2}),$ suppose WLOG that $\mu_{X}=\mu_{Y}=0.$ After long and laborious algebraic manipulations, it is possible to obtain: $f_{X+Y}(z)=\frac{1}{2\pi\sigma_{1}\sigma_{2}}\int_{-\infty}^{+\infty}e^{-\frac{(z-z)^{2}}{2\sigma_{2}^{2}}}e^{-\frac{z^{2}}{2\sigma_{1}^{2}}}\mathrm{d}x=\cdots=\frac{1}{\sqrt{2\pi(\sigma_{1}^{2}+\sigma_{2}^{2})}}\cdot e^{-\frac{z^{2}}{2(\sigma_{1}^{2}+\sigma_{2}^{2})}}.$Therefore $f_{X,Y}$ is the density function corresponding to the distribution $\mathcal{N}(0,\sigma_{X}^{2}+\sigma_{Y}^{2}).$ **Proof by moment generating function**: By [[Moment Generating Function of Normal Distribution]], $M_{X}(t)=\exp\left( \frac{t^{2}\sigma_{X}^{2}}{2} + t \mu_{X} \right),\quad M_{Y}(t)=\exp\left( \frac{t^{2}\sigma_{Y}^{2}}{2} + t \mu_{Y} \right).$ By [[Moment Generating Function of Sum of Two Independent Real-Valued Random Variables]], $\begin{align} M_{X+Y}(t)=M_{X}(t)\cdot M_{Y}(t)&=\exp\left( \frac{t^{2}\sigma_{X}^{2}}{2} + t \mu_{X} \right)\cdot\exp\left( \frac{t^{2}\sigma_{Y}^{2}}{2} + t \mu_{Y} \right) \\ &= \exp \left( (\mu_{X}+\mu_{Y})t + \frac{(\sigma_{X}^{2}+\sigma^{2}_{Y})t^{2}}{2} \right) \end{align}$which shows that $X+Y$ has the same [[Moment generating function of real-valued random variable|moment-generating function]] as $Z\sim N(\mu_{X}+\mu_{Y},\sigma_{X}^{2}+\sigma_{Y}^{2}).$ Therefore, by [[Moment Generating Function of Real-Valued Random Variable Determines Probability Distribution]], $X+Y \sim N(\mu_{X}+\mu_{Y},\sigma_{X}^{2}+\sigma_{Y}^{2}).$